Question

The complete solution of \[ y^3\frac{\partial u}{\partial x} + x^2\frac{\partial u}{\partial y} = 0 \] is \(u(x,y)=\)

• A. \(Ae^{K(x^3-y^4)}\)
• B. \(Ae^{K(x^2+y^3)}\)
• C. \(Ae^{K\frac{x^3}{y^4}}\)
• D. \(Ae^{K\left(\frac{x^3}{3}-\frac{y^4}{4}\right)}\)

Hint:

For a Lagrange PDE, \[ P\frac{\partial u}{\partial x} + Q\frac{\partial u}{\partial y} = R, \] solve the auxiliary equations \[ \boxed{ \frac{dx}{P} = \frac{dy}{Q} = \frac{du}{R}. } \] The complete solution is obtained from the characteristic integrals.

Answer 1

The Correct Option is D